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  "body": "\n## Summary\n\nA Fredholm integral equation is a [[linear integral equation|AYZCKG34QK]] whose integral runs over a fixed set, such as $[a, b]$.\nThe upper limit is not the variable $x$.\nIt comes in a [[first kind|2RMWAVR1HV]], $\\int_a^b K(x, s) \\varphi(s) \\, ds = f(x)$, and a [[second kind|DEYYNVCV9J]], $\\varphi(x) = f(x) + \\lambda \\int_a^b K(x, s) \\varphi(s) \\, ds$.\nFor the second kind with a continuous or square-integrable kernel, [[Fredholm theory|242TFD41KZ]] gives a complete picture of solvability.\nThis picture is much as for square systems of linear equations.\n[[Boundary value problems|TM8VQJWGHZ]] for differential equations lead to Fredholm equations through [[Green's functions|SH6JY9RN2E]].\n\n## Explanations\n\n### L1 human\n\nIn a Fredholm equation, the value of the unknown at each point depends on its values on the whole interval.\nThis is unlike a [[Volterra integral equation|DND8CBEVXT]], where only the values up to the current point matter.\nA Fredholm equation behaves like a very large square system of linear equations.\nFor example, $\\varphi(x) = x + \\int_0^1 x s \\, \\varphi(s) \\, ds$ is a Fredholm equation of the second kind on $[0, 1]$, with solution $\\varphi(x) = 3x/2$.\n",
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